Simplified Successive-Cancellation List Decoding of Polar Codes - - PowerPoint PPT Presentation

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Simplified Successive-Cancellation List Decoding of Polar Codes Seyyed Ali Hashemi , Carlo Condo, Warren J. Gross Department of Electrical and Computer Engineering McGill University Montr eal, Qu ebec, Canada July 12, 2016 Seyyed Ali

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Simplified Successive-Cancellation List Decoding of Polar Codes

Seyyed Ali Hashemi, Carlo Condo, Warren J. Gross

Department of Electrical and Computer Engineering McGill University Montr´ eal, Qu´ ebec, Canada

July 12, 2016

Seyyed Ali Hashemi (McGill) SSCL Decoding of Polar Codes ISIT 2016 1 / 14

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SLIDE 2

Motivation

What is the problem?

5G requirements are stringent

Polar Codes are a good match

Successive-Cancellation List (SCL) Decoding

Very good error-correction performance but high complexity Very slow: there are many redundant calculations

In this talk: We show how to speed up SCL without losing error-correction performance!

Seyyed Ali Hashemi (McGill) SSCL Decoding of Polar Codes ISIT 2016 2 / 14

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SLIDE 3

Background

Polar Codes

Can provably achieve channel capacity Encoding is based on polarizing matrix G⊗n

Input bits are divided into Information bits and Frozen bits Frozen bits help decoding

Decoding schemes:

Successive-Cancellation (SC) SC List (SCL) Sphere Decoding (SD) List-SD

Speed Error-Correction Performance SCL SD SC List-SD

  • E. Arıkan, ”Channel polarization: A method for constructing capacity achieving codes for symmetric binary input

memoryless channels,” IEEE Transactions on Information Theory, vol. 55, no. 7, pp. 3051-3073, July 2009. Seyyed Ali Hashemi (McGill) SSCL Decoding of Polar Codes ISIT 2016 3 / 14

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SLIDE 4

Background

Polar Codes

Can provably achieve channel capacity Encoding is based on polarizing matrix G⊗n

Input bits are divided into Information bits and Frozen bits Frozen bits help decoding

Decoding schemes:

Successive-Cancellation (SC) SC List (SCL) Sphere Decoding (SD) List-SD

Speed Error-Correction Performance SCL SD SC List-SD

  • E. Arıkan, ”Channel polarization: A method for constructing capacity achieving codes for symmetric binary input

memoryless channels,” IEEE Transactions on Information Theory, vol. 55, no. 7, pp. 3051-3073, July 2009. Seyyed Ali Hashemi (McGill) SSCL Decoding of Polar Codes ISIT 2016 3 / 14

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SLIDE 5

Background

Encoding and SC Decoding

u3 u5 u6 u7 x0 x1 x2 x3 x4 x5 x6 x7

  • s = 0s = 1 s = 2

s = 3 s = 3 s = 2 s = 1 s = 0 α β αl βl βr αr

Seyyed Ali Hashemi (McGill) SSCL Decoding of Polar Codes ISIT 2016 4 / 14

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SLIDE 6

Background

Encoding and SC Decoding

u3 u5 u6 u7 x0 x1 x2 x3 x4 x5 x6 x7

  • s = 0s = 1 s = 2

s = 3 s = 3 s = 2 s = 1 s = 0 α β αl βl βr αr

Exact formulation αl

i = ln

1 + eαi+αi+2s−1 eαi + eαi+2s−1

  • αr

i =αi+2s−1 + (1 − 2βl i)αi

Seyyed Ali Hashemi (McGill) SSCL Decoding of Polar Codes ISIT 2016 4 / 14

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SLIDE 7

Background

Encoding and SC Decoding

u3 u5 u6 u7 x0 x1 x2 x3 x4 x5 x6 x7

  • s = 0s = 1 s = 2

s = 3 s = 3 s = 2 s = 1 s = 0 α β αl βl βr αr

Exact formulation αl

i = ln

1 + eαi+αi+2s−1 eαi + eαi+2s−1

  • αr

i =αi+2s−1 + (1 − 2βl i)αi

Hardware-friendly formulation αl

i = sgn(αi) sgn(αi+2s−1) min(|αi|, |αi+2s−1|)

αr

i =αi+2s−1 + (1 − 2βl i)αi

Seyyed Ali Hashemi (McGill) SSCL Decoding of Polar Codes ISIT 2016 4 / 14

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SLIDE 8

Background

SSC and Fast-SSC

SC s = 3 s = 2 s = 1 s = 0 T = 14 time-steps

Seyyed Ali Hashemi (McGill) SSCL Decoding of Polar Codes ISIT 2016 5 / 14

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SLIDE 9

Background

SSC and Fast-SSC

Simplified SC (SSC) s = 3 s = 2 s = 1 s = 0 Rate-0 Rate-1 T = 10 time-steps

Seyyed Ali Hashemi (McGill) SSCL Decoding of Polar Codes ISIT 2016 5 / 14

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Background

SSC and Fast-SSC

Fast-SSC s = 3 s = 2 s = 1 s = 0 Rep SPC T = 2 time-steps

Seyyed Ali Hashemi (McGill) SSCL Decoding of Polar Codes ISIT 2016 5 / 14

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SLIDE 11

Background

SCL Decoding

For finite practical code-lengths, SCL estimates each information bit as either 0 or 1

L codeword candidates survive to limit complexity CRC-aided SCL can outperform LDPC codes

A path metric helps the selection of the surviving candidates

Seyyed Ali Hashemi (McGill) SSCL Decoding of Polar Codes ISIT 2016 6 / 14

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SLIDE 12

Background

SCL Decoding

For finite practical code-lengths, SCL estimates each information bit as either 0 or 1

L codeword candidates survive to limit complexity CRC-aided SCL can outperform LDPC codes

A path metric helps the selection of the surviving candidates Exact formulation PMil =

i

  • j=0

ln

  • 1 + e−(1−2ˆ

ujl )αjl

  • Seyyed Ali Hashemi (McGill)

SSCL Decoding of Polar Codes ISIT 2016 6 / 14

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SLIDE 13

Background

SCL Decoding

For finite practical code-lengths, SCL estimates each information bit as either 0 or 1

L codeword candidates survive to limit complexity CRC-aided SCL can outperform LDPC codes

A path metric helps the selection of the surviving candidates Exact formulation PMil =

i

  • j=0

ln

  • 1 + e−(1−2ˆ

ujl )αjl

  • Hardware-friendly formulation

PMil =

  • PMi−1l +|αil|,

if ˆ uil = 1

2 (1 − sgn (αil)) ,

PMi−1l,

  • therwise

Seyyed Ali Hashemi (McGill) SSCL Decoding of Polar Codes ISIT 2016 6 / 14

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Proposed Algorithm

SCL Issues

SCL requires sorting the path metrics

Adds N more time-steps to the decoding process

SSC and Fast-SSC are equivalent to SC

They are guaranteed to preserve the error-correction performance

Simplified SCL: Faster: simplified Rate-1, Rate-0 and Rep nodes Guaranteed to preserve error-correction performance

Seyyed Ali Hashemi (McGill) SSCL Decoding of Polar Codes ISIT 2016 7 / 14

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Proposed Algorithm

Simplified SCL

Theorem The path metric for a Rate-1 node of length 2s can be calculated as PM2s−1 =

2s−1

  • i=0

ln

  • 1 + e−(1−2βi)αi
  • ,

where αi and βi are relative to the top of the Rate-1 node tree. Based on List-SD idea [α0, α1, α2, α3] ⇔ [β0, β1, β2, β3] s = 2 s = 1 s = 0

Seyyed Ali Hashemi (McGill) SSCL Decoding of Polar Codes ISIT 2016 8 / 14

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SLIDE 16

Proposed Algorithm

Simplified SCL

Proof. Induction (ηi = 1 − 2βi) For Rate-1 node with N = 2:

SC            αl

0 = ln

  • 1+eα0+α1

eα0+eα1

  • αr

0 = α1 + ηl 0α0

ηl

0 = η0η1

ηr

0 = η1

PM1 = ln

  • 1 + e−ηl

0αl

  • +ln
  • 1 + e−ηr

0αr

  • Substituting for αl

0, αr 0, ηl 0 and ηr 0:

PM1 = ln

  • 1 + e−η0α0

+ln

  • 1 + e−η1α1

The theorem holds for N = 2.

Seyyed Ali Hashemi (McGill) SSCL Decoding of Polar Codes ISIT 2016 9 / 14

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SLIDE 17

Proposed Algorithm

Simplified SCL

Proof. Induction (ηi = 1 − 2βi) For Rate-1 node with N = 2:

SC            αl

0 = ln

  • 1+eα0+α1

eα0+eα1

  • αr

0 = α1 + ηl 0α0

ηl

0 = η0η1

ηr

0 = η1

PM1 = ln

  • 1 + e−ηl

0αl

  • +ln
  • 1 + e−ηr

0αr

  • Substituting for αl

0, αr 0, ηl 0 and ηr 0:

PM1 = ln

  • 1 + e−η0α0

+ln

  • 1 + e−η1α1

The theorem holds for N = 2. Proof. For Rate-1 node with N = 2s:

PM2s−1 = PMl

2s−1−1 + PMr 2s−1−1

= Σ2s−1−1

i=0

ln

  • 1 + e−ηiαi

+ ln

  • 1 + e−ηi+2s−1 αi+2s−1

= Σ2s−1

i=0 ln

  • 1 + e−ηiαi

,

The theorem holds for all N.

u0 u1 u2 u3 β0 β1 β2 β3 x0 x1 x2 x3

Seyyed Ali Hashemi (McGill) SSCL Decoding of Polar Codes ISIT 2016 9 / 14

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Proposed Algorithm

Simplified SCL

Theorem In the hardware-friendly formulation of SCL algorithm, the path metric for a Rate-1 node of length 2s can be calculated as PM2s−1 = 1 2

2s−1

  • i=0

sgn (αi) αi − (1 − 2βi)αi, where αi and βi are relative to the top of the Rate-1 node tree. Simplified Rate-1 node can be decoded in N time-steps

About a factor of 3 faster than conventional SCL

Seyyed Ali Hashemi (McGill) SSCL Decoding of Polar Codes ISIT 2016 10 / 14

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Proposed Algorithm

Simplified SCL

Theorem The path metric for a Rate-0 node of length 2s can be calculated as PM2s−1 =

2s−1

  • i=0

ln

  • 1 + e−αi

, where αi is the LLR value at the top of the Rate-0 node tree. [α0, α1, α2, α3] ⇔ [0, 0, 0, 0] s = 2 s = 1 s = 0 Simplified Rate-0 nodes can be decoded in at most log2 N time-steps

The time complexity is reduced from O(N) to O(log2 N)

Seyyed Ali Hashemi (McGill) SSCL Decoding of Polar Codes ISIT 2016 11 / 14

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Proposed Algorithm

Simplified SCL

Theorem The path metric for a Rep node of length 2s can be calculated as PM2s−1 =

2s−1

  • i=0

ln

  • 1 + e−(1−2β2s −1)αi
  • ,

where αi is relative to the top of the Rep node tree and β2s−1 is relative to the information bit in the Rep node tree. [α0, α1, α2, α3] ⇔ [β3, β3, β3, β3] s = 2 s = 1 s = 0 Simplified Rep nodes can be decoded in at most log2 N time-steps

The time complexity is reduced from O(N) to O(log2 N)

Seyyed Ali Hashemi (McGill) SSCL Decoding of Polar Codes ISIT 2016 12 / 14

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Conclusion

Conclusion

We proposed Simplified SCL

Faster decoding of Rate-1, Rate-0 and Rep nodes non-approximated formulation

Guaranteed equivalence with conventional SCL

Polar code of length 2048 and rate 1

2

Conventional SCL decoding requires 6142 time-steps Simplified SCL decoding requires 1741 time-steps More than a factor of 3.5× speed-up

Ongoing research: extending the technique to SPC nodes and implement it on hardware

Seyyed Ali Hashemi (McGill) SSCL Decoding of Polar Codes ISIT 2016 13 / 14

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Conclusion

Thank you!

Seyyed Ali Hashemi (McGill) SSCL Decoding of Polar Codes ISIT 2016 14 / 14